Scalar product and perpendicular vectorsAQA A-Level Further Maths: Subtopic test

1. 1
   The vectors $\mathbf a=\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and $\mathbf b=\begin{pmatrix} 4 \\ 5 \\ 1 \end{pmatrix}$ are given.

   (a)

   Find $\mathbf a\cdot\mathbf b$.

   [1 mark]

   • A$\begin{pmatrix} 8 \\ -5 \\ 3 \end{pmatrix}$
   • B$16$
   • C$6$
   • D$0$

   (b)

   Which of these vectors is perpendicular to $\mathbf a$?

   [1 mark]

   • A$\begin{pmatrix} 1 \\ -2 \\ 0 \end{pmatrix}$
   • B$\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}$
   • C$\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$
   • D$\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$

   (c)

   Find the angle between $\mathbf a$ and $\mathbf b$, giving your answer to the nearest $0.1^\circ$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Triangle $PQR$ has vertices $P(1,2,3)$, $Q(4,0,5)$ and $R(3,4,2)$.

   (a)

   Find $\overrightarrow{PQ}\cdot\overrightarrow{PR}$.

   [1 mark]

   • A$12$
   • B$4$
   • C$8$
   • D$0$

   (b)

   What does this tell you about triangle $PQR$?

   [1 mark]

   • AIt has a right angle at $P$.
   • BIt has a right angle at $Q$.
   • CIt has a right angle at $R$.
   • DIt has no right angle.

   (c)

   Hence find the exact area of triangle $PQR$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The lines $l_1$ and $l_2$ have equations $\mathbf r=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+\lambda\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}$ and $\mathbf r=\begin{pmatrix} 3 \\ -1 \\ 0 \end{pmatrix}+\mu\begin{pmatrix} 1 \\ 4 \\ 8 \end{pmatrix}$.

   (a)

   Find the acute angle between $l_1$ and $l_2$, to the nearest $0.1^\circ$.

   [3 marks]

   (b)

   The line $l_3$ has direction $(k,1,2k)$ and is perpendicular to $l_1$. Find $k$, and then find the angle between $l_3$ and $l_2$, to the nearest $0.1^\circ$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A vertical mast stands on horizontal ground with its foot at the origin $O$. A support wire runs in a straight line from the top of the mast, $A(0,0,12)$, to a point $B(3,4,0)$ on the ground. Distances are in metres.

   (a)

   (i) Show that the length of the wire is $13$ m.
   (ii) Find the angle between the wire and the mast, to the nearest $0.1^\circ$.

   [6 marks]

   (b)

   A straight walkway runs along the ground from $B$ to a point $C(c,0,0)$ on the $x$-axis, so that $BC$ is perpendicular to $OB$.
   (i) Find the value of $c$.
   (ii) Hence find the area of triangle $OBC$.

   [6 marks]

   Total for question 4: 12 marks

End of questions